Inspired by [1]} we propose here a mixed-binary and real-valued version of the aforementioned model,
by this meaning that while we assume that the agent's opinions take (positive or negative) real values, that represent their levels of appreciation or
dislike of each specific issue, we do not quantify the level of mutual appraisal, but only take into account whether the mutual relationships between pairs of agents are friendly or hostile.
This simplified assumption has been adopted in our previous work [2]}, as well as in the works of Cisneros-Velarde et al. [3]} and of Mei et al. [4]} in which mutual appraisals are treated as binary variables.
In particular, in [4]} the evolution of a signed unweighted non-all-to-all network towards the straightforward generalization of the concept of structural balance has been proposed, thus leading to the graph-theoretical concept of “triad-wise structural balance", where each agent's ego-networks satisfies the structural balance property.
This discrete-time “gossip-like" model enjoys the property of convergence towards a non-all-to-all structural balance configuration, while the structure of the associated graph is time invariant, since only the signs of the weights change. In [3]} a network formation game, in which pairs of rational individuals strategically change the signs of the edges in a complete network is proposed. The game is shown to strategically reduce the cognitive dissonance in the network along time, by driving the network towards clustering balance [7]}.
The motivation behind the study of the dynamical evolution of unweighted signed social networks, that units the aforementioned works, comes from the fact that there are many circumstances in which recognizing the type (friendly or hostile) of relationship between individuals is easy, while assessing its intensity
is complicated and prone to model errors.
In fact, while individual evaluations of certain products or their opinions on certain topics can be easily obtained, attributing numerical values to the
mutual appraisals is more challenging and oftentimes individuals prefer to not even reveal them.
